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Properties

Label 1-6017-6017.1667-r0-0-0
Degree $1$
Conductor $6017$
Sign $0.440 + 0.897i$
Analytic cond. $27.9428$
Root an. cond. $27.9428$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.892 + 0.450i)2-s + (−0.309 − 0.951i)3-s + (0.594 − 0.804i)4-s + (−0.369 − 0.929i)5-s + (0.704 + 0.709i)6-s + (−0.932 + 0.362i)7-s + (−0.168 + 0.985i)8-s + (−0.809 + 0.587i)9-s + (0.748 + 0.663i)10-s + (−0.948 − 0.316i)12-s + (0.104 + 0.994i)13-s + (0.669 − 0.743i)14-s + (−0.769 + 0.638i)15-s + (−0.293 − 0.955i)16-s + (−0.984 + 0.176i)17-s + (0.457 − 0.889i)18-s + ⋯
L(s)  = 1  + (−0.892 + 0.450i)2-s + (−0.309 − 0.951i)3-s + (0.594 − 0.804i)4-s + (−0.369 − 0.929i)5-s + (0.704 + 0.709i)6-s + (−0.932 + 0.362i)7-s + (−0.168 + 0.985i)8-s + (−0.809 + 0.587i)9-s + (0.748 + 0.663i)10-s + (−0.948 − 0.316i)12-s + (0.104 + 0.994i)13-s + (0.669 − 0.743i)14-s + (−0.769 + 0.638i)15-s + (−0.293 − 0.955i)16-s + (−0.984 + 0.176i)17-s + (0.457 − 0.889i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.440 + 0.897i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.440 + 0.897i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6017\)    =    \(11 \cdot 547\)
Sign: $0.440 + 0.897i$
Analytic conductor: \(27.9428\)
Root analytic conductor: \(27.9428\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6017} (1667, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6017,\ (0:\ ),\ 0.440 + 0.897i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.006111432390 + 0.003809248541i\)
\(L(\frac12)\) \(\approx\) \(0.006111432390 + 0.003809248541i\)
\(L(1)\) \(\approx\) \(0.3872768512 - 0.1341628158i\)
\(L(1)\) \(\approx\) \(0.3872768512 - 0.1341628158i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
547 \( 1 \)
good2 \( 1 + (-0.892 + 0.450i)T \)
3 \( 1 + (-0.309 - 0.951i)T \)
5 \( 1 + (-0.369 - 0.929i)T \)
7 \( 1 + (-0.932 + 0.362i)T \)
13 \( 1 + (0.104 + 0.994i)T \)
17 \( 1 + (-0.984 + 0.176i)T \)
19 \( 1 + (-0.818 - 0.574i)T \)
23 \( 1 + (0.996 + 0.0804i)T \)
29 \( 1 + (-0.958 + 0.285i)T \)
31 \( 1 + (-0.485 - 0.873i)T \)
37 \( 1 + (0.231 - 0.972i)T \)
41 \( 1 + (-0.978 - 0.207i)T \)
43 \( 1 + (-0.200 - 0.979i)T \)
47 \( 1 + (0.991 + 0.128i)T \)
53 \( 1 + (-0.554 - 0.832i)T \)
59 \( 1 + (-0.818 + 0.574i)T \)
61 \( 1 + (-0.704 - 0.709i)T \)
67 \( 1 + (-0.200 - 0.979i)T \)
71 \( 1 + (-0.513 - 0.857i)T \)
73 \( 1 + (-0.247 - 0.968i)T \)
79 \( 1 + (0.215 - 0.976i)T \)
83 \( 1 + (0.619 - 0.784i)T \)
89 \( 1 + (0.354 - 0.935i)T \)
97 \( 1 + (-0.789 - 0.613i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−17.52118029732091535862417140825, −16.98221684072205055286591360657, −16.4023155150390591680905072872, −15.6056077930640376725216235059, −15.30815611070375778265552469134, −14.64167264236535135091524216118, −13.50656335195679530960351076408, −12.85822667818475487720979133458, −12.10624872530177864027769029710, −11.32762666435449786005958029, −10.65834775342739729228282543324, −10.53310320223394887616629694548, −9.72927948056738842942711014241, −9.13905265395900736968907636967, −8.37361653869370924875078236201, −7.62827720407198335269592646119, −6.7082486920203160667646061559, −6.42368428174584967369777276861, −5.43085901891640539328047324716, −4.23295871030454264372036411893, −3.75370900230987433649335732346, −2.940561011499684753113128659588, −2.65774165261996298539781140930, −1.220851190519293976790974505149, −0.0052623536626390449022837237, 0.47530033140066330691512916530, 1.79420264860441271254189124383, 2.00731073127042414946821908155, 3.16723582167006745866913438815, 4.33050733462379477890637470963, 5.148002822293348341804774805536, 5.912031451673487931853859715271, 6.48383192676517929302745721174, 7.121483693241282952568879261899, 7.661941176615353447495309939483, 8.69537873538569404862884240722, 8.98144108232361260228299872912, 9.42532902340272207636851202110, 10.72504264993309690665536941651, 11.13472009060866796268350482665, 11.92982129525209667485243109244, 12.48656317121320508936717674788, 13.28622569412298126740168304743, 13.62128962572596138239187833552, 14.847169480368155927506676574158, 15.33742016756048952260906538476, 16.12092783240797135324191981580, 16.73312651716285241949242263070, 17.030082554554905236579731434576, 17.74378500058707979781466568711

Graph of the $Z$-function along the critical line